Emma waited for Jacob’s mask of bemusement to give way to the hint of wounded pride that she’d been aiming for. They both knew that she was speaking the truth. Jacob was nimble enough — even graceful at times — but he just didn’t put in the hours in the gym. No one would vote to grant permanent immunity to a dancer with a pudgy, childlike belly and toneless upper arms.

“It’s not that we didn’t have fun,” she continued, glancing across Briggs Field, wondering if any of her followers had come to witness the break-up in person. “But we’re headed for different fates, and it’s better to acknowledge that now than spend years together fooling ourselves that it could ever work out in the end.”

Jacob took a moment to gather his thoughts. “I’m sure you’ll get tenure yourself,” he said. “You might be a one-shtick pony, but for stand-up that’s all they require.”

“Ouch. If that’s the best heckle you’ve got, I wouldn’t advise switching to comedy.”

“Goodbye, Emma.” Jacob turned and walked away.

Emma watched his receding figure, annoyed; she’d expected him to make much more of a scene. Her PR app had been nagging her for weeks to break up with him, and it had shown her convincing projections for retentions and new followers if she managed to get some fireworks out of the event. But she’d pulled her punches: she hadn’t goaded him anywhere near enough.

“The sex was terrible!” she shouted after him. “I’ve had more orgasms from my ringtone!”

He didn’t even turn to acknowledge the line. Emma felt a strange hollowness, but then she raised her right hand and offered her followers an air high-five. A satisfying thwack sounded in her earpiece, and her glasses showed the response: six thousand and twelve, out of the not-quite ten thousand taking the stream. Not great, but not a disaster. She sampled her tweets, sorting them so the positives came first: U rock, grrl! Really wiped the grass with that loser!

For five minutes Emma stood on the path beside the playing field, checking half a dozen indicators to gauge the success of the move. All in all she’d come out in front, but she wasn’t satisfied.

She could hardly go chasing after Jacob now, in the hope of provoking a stronger reaction; she’d just seem desperate. But the fact that he hadn’t betrayed much emotion to her face didn’t mean he wasn’t angrier or more cut-up than he’d appeared.

Emma gestured for her bookmarks and scrolled down to CrowdStalker.com. CrowdStalker’s coverage had been patchy in Des Moines, but Cambridge had more citizen surveillance than any other city on Earth. She uploaded a selection of pictures of her ex, and endured the three-minute Pepsi ad necessary to pay for the service.

Job done, she cleared her frames and spent a moment taking in the ambience. The path that ran past Simmons Hall was packed with her fellow poopy-heads, striding along with one eye on their feeds, ambition fairly oozing from their pores. Competition among the physics students for followers was ruthless — in stand-up more than any other mode — but with Spring Term starting tomorrow at least she had clear air around her and nothing holding her back. She couldn’t afford to be sentimental, and in the end Jacob would thank her. He’d never soar as high as she would, and the sooner he understood that the less pain he’d have to suffer coming to terms with the truth.

Emma was in her Tuesday morning improv workshop when her glasses flagged the CrowdStalker report. She finished her vocal warmup then excused herself, miming stomach cramps.

In a bathroom stall she scanned the report, annoyed that it hadn’t reached her sooner. The night before, Jacob had been seen entering a community hall in Boston: a small, shabby building that looked like it might host AA meetings for cockroaches. But the passing pair of digital eyes on the lookout for CrowdStalker targets had also captured the temporary sign taped to the door: STUDY PHYSICS IN CAIRO.

Cairo? Emma’s irritation melted away. She’d broken Jacob’s heart so badly that after a fortnight without her he was ready to flee the country — like some tortured hero in a nineteenth-century novel, joining the Foreign Legion to escape the aftermath of a tragic, doomed affair.

But what were his chances of making it to Egypt? A little googling revealed that the scholarship program was only offering ten places for the entire country. Maybe his subdued performance style would be more appealing to an Egyptian judge than an American, but Emma seriously doubted that he was top-ten material by anyone’s standards.

Either way, one shot of Jacob outside the hall wouldn’t make for much of a blog post. But the applications were continuing for another two nights, and a few scenes from the auditions might give her something to work with. If she applied for the scholarship herself she could get into the hall without any awkward questions, and the next best thing to having video of Jacob pouring out his soul on some creaky wooden stage would be to catch another failed poopy-head who couldn’t handle the main game, desperately cavorting before a panel of sheikhs in the hope of a second chance in the minor league.

Emma could hear the workshop getting underway, and it sounded like Katherine Bremmer was hogging the scene, as usual. She told her butler to book a place for her among the wannabe legionnaires, and hurried back to join the group before they all forgot that she existed.

“Dance, rap or stand-up?” Emma asked the slender girl in front of her in the queue. It was a joke: dance, obviously.

“Physics,” the girl replied.

“Er—” Emma gestured at the sign on the door. “Obviously. But what mode?”

“That’s not how they do things over there.” The skinny ballerina regarded Emma with amusement. “Did you think ... ?”

Emma smiled and shook her head. “No, I’m just messing with you.” She googled desperately for a few seconds, but nothing that came up among the first few hits told her what she needed to know. “I understand that it’s all ... a kind of freestyle,” she bluffed.

“I suppose that’s one way of putting it,” Ballet Bitch replied.

Emma had no interest in the prize on offer here, but if she was unprepared for the competition it would still be humiliating. “Do you think I’ll need a change of clothes?” she asked, spreading her arms to allow the woman to make a ruthless assessment of her jeans and sweatshirt. Most of the poopy-heads in the queue were carrying backpacks, so for all she knew they might have been toting belly-dancing costumes or billowing Arabian Nights trousers.

“Ha!” Emma felt her cheeks flush slightly, and she waited for the inevitable follow-through, but instead of raising a hand to invite her followers to share in the afterglow, the woman just turned away.

When the doors opened and the queue started moving, Emma saw a sign ahead in the foyer that the CrowdStalker snap hadn’t revealed: a WiFi logo behind a barred red circle, and a warning that any attempts at net access or recording would be jammed. She thought of turning around and walking away: if she couldn’t get this on video, what was the point? But the line was moving forward — and she’d promised her followers an update at ten. All she could do was observe this off-Broadway farce for herself, and then improvise a re-enactment of the night’s low-lights. It would be challenging, but no one could say she wasn’t an innovator.

In the foyer, a woman in her thirties was handing pads of paper and pens to the people filing past. Emma accepted the entrance kit, then flipped open the cover of the pad. The first page was blank ... and so were all the rest. No guidance here. Her stupid butler should have gleaned everything important from the application web page and given her a heads up, but it was too late to berate it now and ask for the details; it lived in the Cloud, and her glasses had already lost access.

There were a couple of dozen chairs set up in the hall, and some kind of projector and screen at the front. Emma took a seat at the end of the back row and waited for things to become clear, trying not to let her body language betray any anxiety to the poopy-head beside her.

The woman who’d handed out the pads walked to the front of the hall. “Welcome! My name’s Ghada.” The space was small enough that she could make herself heard without any need for a microphone. “We have nineteen applicants tonight, so I’m going to have to impose the ten minute limit very strictly. If you need to leave before everyone’s done, please do it quietly. I expect some of you are nervous, but don’t worry, I’m a very gentle interrogator. Good luck to everyone, and let’s get started.” She consulted a sheet on her clipboard. “Robert Lopez.”

A young man rose to his feet. As he was making his way forward, Ghada switched on the projector and a diagram appeared on the screen: a simple circuit comprising a few resistors, capacitors and coils. “I apply a fifty-Hertz voltage source between terminals A and B,” she said. “What can you tell me about the relative size and phase of the voltage that will appear between points C and D?”

Lopez began scribbling on his pad. For three or four minutes the entire hall was silent, with not so much as a nervous cough — though some members of the audience were busy making their own notes. Emma was tempted to start jeering, but she contained herself: perhaps they were playing by different rules here. Maybe Lopez had five minutes to write his lyrics, choreograph his moves, or whatever, with the actual performance only coming in the second half of his allotted time.

“I’d say the voltage there is half the applied value,” he announced. “And it leads it by forty-five degrees.”

“OK,” Ghada said noncommittally. “So what’s your argument?”

Lopez leaned down and put his pad under the projector’s lens, then he began describing the steps that had led him to the answer. Emma let his words wash over her and just gazed in astonishment: he had turned his back to the audience. Done provocatively that could be a powerful move — but it looked more like he’d simply forgotten that they were there.

A woman in the second row raised her hand. “Can you explain how you get the impedance for those two coils in parallel?”

“Sure.” Lopez expounded on the details in boyishly enthusiastic tones. It sounded like he knew his stuff, but Emma’s skin crawled with pity and embarrassment. It wasn’t what a poopy-head knew that counted. Delivery was everything.

The competitors that followed him were much the same: technically competent, but none of them had Game. Emma thought of walking out before her time came, lest she end up with an unwanted seat on a plane. But then, maybe she could make some deal where Jacob took her place. She summoned her PR app for its opinion on that strategy, discreetly air-typing in the aisle beside her, but most of its smarts were in the Cloud and all it could do was promise to get back to her.

“Emma Stone.”

Emma strode to the front of the room. “Good to see you all in beautiful Boston,” she quipped. “Sicut patribus sit deus nobis! But you really didn’t need to come this far. I could have hooked you up with a crack dealer right on campus.”

The response from her fellow competitors was frosty, but it was the judge who mattered. Emma turned to Ghada, who appeared as nonplussed as if she’d just been served prime steak after thirteen courses of stale cheeseburger. “If we can concentrate on the question,” Ghada finally managed, “time’s ticking away.”

“Of course.” Emma looked around at the screen.

“Suppose a planet is moving in a circular orbit,” Ghada began. “The arrows here are its velocity vectors at different points on its trajectory. If we gather them together the result will look like this.” She put up a second diagram. “For a circular orbit, the velocity vectors themselves form a circle.”

Emma said, “I get it.”

Ghada placed a third sheet under the lens.

“The question is: for an elliptical orbit, what shape do the velocity vectors form? I’m not expecting a rigorous proof, though you’re welcome to give one if you can. If not, just state your conjecture about the shape and then test it: check a few points that could falsify your claim if it’s not correct.”

Emma stared at the orbit. She usually had planet jokes on tap, but they’d all deserted her. “An ellipse,” she said. “A circle gives a circle, an ellipse gives an ellipse. Try saying that ten times quickly after twenty Margaritas.”

Still nothing from the corpses in the audience, but Emma could have sworn she’d brought a smile to Ghada’s elliptical lips.

“Can you test that?” Ghada asked her.

Emma felt her mouth growing dry. Calculations weren’t really her strong suit; she was more of a Big Picture poopy-head. But once she raised the blank pad and began, she found her rhythm. It was just a matter of dividing the planet’s angular momentum by its mass and the appropriate distance, for a few points around the orbit.

“This is what I get,” she said, placing the result under the lens. “It looks pretty elliptical to me.”

Ghada laughed. “Fine. But don’t give up now; if the algebra for the general case seems too daunting, you can test your conjecture with specific numbers. That won’t prove it — but a single counter-example would still disprove it.”

Emma forced herself to concentrate. “Set the mass and the angular momentum both to one,” she said. “And then for the distances ... ” She stared at the annoying fractions and square roots. “One-eighth and one-eighteenth. So the velocities are eight, eighteen and twelve.”

“Good choice,” Ghada said encouragingly. “Now draw in the foci of your ellipse, and see what else you can determine.”

Emma redrew the diagram with the numbers she’d chosen, and figured out where the second focus of the ellipse would have to be, by symmetry. The sum of the distances from the foci needed to be the same for any point on the ellipse, and for the bottom-most and top-most points that sum was twenty-six. For the point she’d found on the side, the sum was twelve from one focus ... plus the square root of two hundred and forty-four from the other.

So either she’d made a mistake with the arithmetic, or the velocities did not form an ellipse at all. She looked up from the pad; Ghada was watching her expectantly.

“I call wilful abstraction!” Emma proclaimed, raising her right hand out of habit and slapping the air for a dose of affirmation that never came.

“I’m sorry?” Ghada was perplexed. Apparently they hadn’t heard of that rule in Egypt.

“You think this has no applications?” Ghada was amused. “It might strike you that way, but it’s not the case. Describing the figure that these velocities form is one route to a deeper understanding of any inverse-square force — including the electrostatic force in an atom. With a bit more work, this problem offers a short cut to the energy levels of hydrogen. So in a sense it underlies the first approximation we have to the properties of every single atom, the structure of the periodic table, and a rough stab at making sense of all of chemistry. Every rock, every leaf, every molecule in your body. Is that still too abstract for you?”

Before Emma could reply, Ballet Bitch called out, “The phrase comes from Norman Reece. He was a vice-presidential candidate ten years ago, but he had to pull out of the race when someone dug up what he’d tweeted after failing Physics 110. That’s when he quit his engineering degree and went into law.”

“Go on,” Ghada urged her. “You can’t offer us that much political intrigue and then cut the story short before we’ve heard the tweet itself.”

Ballet Bitch shifted uncomfortably in her chair, but then replied, “‘Quantum mechanics is just a wilful abstraction, dreamed up by Jews and faggots to try to distract us from the real world.’”

“Much worse,” Ghada concurred. “But the bigotry here is far less interesting than the other flaws driving it. Here’s a young man who embarked on a career that required a basic understanding of the world — one that we’ve possessed for more than a century, and that millions of people have managed to come to grips with. But when he failed in that modest ambition, all he could do was look for someone else to blame. And I suppose that’s what the whole ‘wilful abstraction’ ploy is about: an attempt to reframe incompetence as a kind of laudable practicality—”

Emma headed for the door. She didn’t have to put up with this shit. Ghada called after her, but Emma wasn’t listening; she marched out onto the street, her face burning, blood pounding in her ears.

She was halfway to the bus stop when she remembered that she’d promised her followers pictures at ten. Even if she told them that she’d stormed out after being insulted, they’d want more than a face-to-webcam monologue from her room.

Reluctantly, Emma walked back to the hall. She placed her watch on a trashcan and captured a few takes re-enacting and explaining her exit. None of them felt great. She decided to hang around until the meeting broke up, so she could get some footage of all the losers leaving the building. She could already picture the sarcastic captions, in faux-scratched-celluloid style, following each figure as they headed into the night.

There was a thrift shop on the opposite side of the street, closed at this hour but with security lighting showing through the windows. Emma found an unsoiled patch of ground beside the door and hunkered down to wait. She still had the pad Ghada had given her, and after a few minutes she gave in to the temptation to start checking her arithmetic to see where she’d gone wrong.

But she hadn’t made a mistake with the calculations. The points simply didn’t lie on an ellipse — or at least, not one with the focus she’d been assuming.

She played with the numbers some more, trying out different possibilities, and then suddenly everything was clear.

The tips of the velocity vectors lay on a perfect circle. It just happened to be a circle whose centre was offset from the origin of the vectors, allowing them to take on different lengths.

Emma groaned in self-reproach. If she’d kept her cool and worked this out on stage, she could have won the scholarship and then thrown it back in that stupid woman’s face.

The losers were filing out of the hall. Emma scrambled to her feet and zoomed her spec-cam. It didn’t really matter how they looked; she could spin the footage any way she wanted.

“I suppose I feel like I’m repaying a debt. My mother studied physics in America — in the glory days, when that meant something. Students came here from all around the world, often from countries deep in crisis. Not many Americans can make it to somewhere really prestigious, like Shanghai, so it only seems right to try to offer an alternative.”

Glory days? Emma was indignant. “You do know that twenty million people live-streamed the makeover episode of American Poopy-head?”

“Yeah, that sums it up.” Ghada laughed sadly. “The only way for a scientist to be halfway palatable in your culture now is through a kind of ritual self-abasement: take on a crass, childish insult as a label and then join in a freak show where they might as well be biting the heads off chickens.”

“You think ‘poopy-head’ is an insult?” The woman’s English was excellent, but she obviously hadn’t mastered the vernacular. “It’s a compliment,” Emma explained. “Sort of a safe-for-work way of saying that someone ‘knows shit’ — that they ‘know stuff’.”

Ghada thought for a moment. “When I was seven, Ahmad Al-Masri told me I was a snot-face, after he came across me proving Pythagoras’s Theorem to some of my friends.”

“See!” Emma replied. “It’s the same everywhere. We’re sisters under the skin.”

“Er, not from that we aren’t. I asked my mother what I should do, and she said, ‘The boy’s still an infant, cut him some slack.’ But if he’d still been talking that way a year later, I would have told him in no uncertain terms what a sad excuse for a human being he was. I wouldn’t have started pandering to him by replying, ‘Yes, I am a proud snot-face!’ I wouldn’t have put it on a T-shirt.”

They’d reached the diner. Emma followed Ghada to a corner booth, and they ordered coffee and cake.

“Every profession has a nick-name,” Emma argued.

“Maybe, but no one calls the Surgeon General here the ‘Quack-in-Chief’, or the Attorney General the ‘Shyster-in-Chief’. So why does your President refer to her Nobel-prize-winning Energy Secretary as ‘Poopy-head-in-Chief’?”

“If that’s too undignified for you,” Emma asked irritably, “what do they call people who do science in Egypt? ‘Masters of the Universe’? ‘Philosopher Kings’?”

“‘Scientists’,” Ghada replied. “Literally. We’ve taken to using the English word, since it seems to have dropped out of favour in the West. ‘Geek’, ‘nerd’, ‘poopy-head’, ‘snot-face’ ... these aren’t words in any adult’s vocabulary. To use them at all is a concise confession by the speaker that, linguistically, they’ve never left kindergarten — but it’s only in the most damaged cultures that people are required to pretend that they’re anything other than infantile jibes. Every time you answer to a label like that, you’re just normalising and internalising your society’s pathological anti-intellectualism.”

Emma bristled. “So now the country you claim to be indebted to is ‘pathological’?”

“It wasn’t always this way,” Ghada stressed. “But it’s a long, sad fall from The Feminine Mystique and The Pleasure of Finding Things Out to The Poopy-head Manifesto and Yes, I Have Girl Cooties, You Wanna Make Something Of It?”

Emma hadn’t heard of the first two books, but Google knew her well enough to track down the authors and then send her to SmugClub.com. “Richard Feynman spent time in topless bars. Betty Friedan never acknowledged her white, middle-class heterosexual privilege. Some heroes you’ve got.”

“I have no heroes,” Ghada said flatly. “But I can recognise a culture in decline when I see it. America is now what anthropologists call a Kardashian Type Three civilisation: more than fifty percent of GDP is in the attention economy.”

“And it’s ‘Grrl Cooties’, not ‘Girl Cooties’. G-double-R-L. When we spell it that way it makes us powerful.”

Ghada seemed to be struggling not to burst out laughing. “God help us all. So why did you apply for the scholarship, if you’re so deliriously happy here?”

Ghada no longer looked amused. “It’s your friends who are lamenting your decline. Your enemies are the ones standing back in silence, waiting for you to choke to death on your own vomit.”

The waitress placed two plates of chocolate cake in front of them, then sensed the tension and withdrew without a word.

Emma broke the silence. “It’s a circle,” she said. “The velocities form a circle. I figured that out.”

“Good.”

She was still angry, but her curiosity got the better of her. “Does that really have something to do with energy levels in atoms? Or were you just bluffing?”

Ghada took a sheet of paper from her clipboard and made a quick sketch.

“Suppose you grab hold of a planet at either point on its orbit when it’s moving parallel to the axis of the ellipse. Swing it around on a circular arc centred on the sun, by any angle you like, but leave its velocity vector exactly the same. Since you’re not changing its speed or its distance from the sun, you’re not changing its kinetic or potential energy, so its total energy remains the same.”

She turned the paper over and drew three circles.

“If the total energy remains the same,” Ghada continued, “the velocity when the planet’s moving parallel to the axis is always the same — which means that all the circles intersect like this, at the same two points.”

“I get it,” Emma replied. “But I still don’t see any atoms.”

Ghada produced another sheet and embarked on a third drawing. Emma looked on, sipping her coffee, sceptical but not quite impatient enough to interrupt.

“Those circles of different sizes that the velocities produce,” Ghada said, “are the projections of great circles from a sphere that sits on the plane. You pass a line from the north pole through a point on the sphere, and where it hits the plane you get a point on one of the velocity circles. It’s the same form of projection they use in some maps.”

“All right.” Emma understood the construction, but she still didn’t see what it was meant to imply.

“The transformation between orbits with the same energy is just a rotation on that sphere,” Ghada claimed. “And when you account for all the possible ways that an ellipse can be oriented in three dimensions, you end up with the fact that all the orbits with a given energy are connected to each other by rotations in four dimensions.”

Emma felt a tingling on the back of her neck. Squeezing an orbit into a skinnier ellipse did not seem to be the same thing as rotating it — but this showed that, in a sense, it was.

Ghada said, “What does this say about a hydrogen atom? If you can rotate a system without changing its energy, in quantum mechanics you can describe it with a few simple numbers that characterise its spin. From there, it’s just a straightforward calculation to find the energy levels.” She slid the paper to one side and plunged a fork into her slice of cake.

On the bus back to Cambridge, Emma sat reviewing her footage, but there was nothing she could use. So much for live-blogging the legionnaires; maybe she should simply claim that the event had been cancelled.

She gazed down at the drawings that Ghada had left her. As a child, Emma had believed that in time she’d learn all the secrets of the world: the rules underlying everything from the farthest galaxy to the cells of her own body. But somewhere along the way she’d discovered that her curiosity was a joke, a sickness, and that the only way to admit to it was to mock it more fiercely than any would-be detractor. And then the wind had changed and that studied pose had frozen in place. She’d let the people who despised everything she’d loved crawl so far into her head that she could no longer tell herself apart from them.

Her PR app began buzzing in her ear, and her glasses showed her the numbers. If she didn’t give her followers something snappy soon, she was going to start losing people — and if she dropped below ten thousand she’d be nothing.

Emma raised her wrist, framing her face squarely with her watch’s camera. “Ever wonder what happens to wannabe poopy-heads who can’t hack it in the real world? Tonight I was thrown out of a hall in Boston by thugs who don’t want you to learn the answer, but your intrepid reporter survived to bring you all the goss.”

She finished the performance just as the bus approached her stop. She rose, leaving the diagrams behind. The last thing she wanted to carry with her was a reminder of everything she’d lost.

Read more on the four-dimensional symmetry of the inverse-square force.